From that article..."On a website called thriftyfun.com, thousands of users posted tip after penny-saving tip on conserving cash. There wasn't a crisis on earth, it seemed, that couldn't be averted with some combination of baking soda, white vinegar, lemon juice, salt, and a certain dishwashing liquid. I squirted a shot of Dawn in a kettle of boiling water, poured it into the shower, and the goop slrrrrged down the drain. Materials used: 10 cents. The look on Ruth's face after I actually fixed something: priceless."

Baking soda and vinegar react together to produce Carbon dioxide and salt water. They do create a vigorous bubbling effect, but this on its own doesn't do any cleaning, and I wouldn't call it explosive. It is also used by kids to create fake volcanoes for science class, and is absolutely benign. Baking soda on its own, with boiling water, could be a good drain cleaner as most clogs are grease, and baking soda, which is a base, reacts with grease to create soap. The hot water will speed this reaction up, dissolve the soda of course, and flush away the soap.

A bathroom drain may very likely be clogged with hair. Try removing the cover and seeing if that is the case. You can just pluck the wad of gucky hair out. If this is your clog, you will not do anything with baking soda or vinegar, alone or together. Drano, or some other brand, will get rid of it, but it is more effective to manually pluck it out.